Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
ALGEBRA II LESSON 5-6 RADICAL EXPRESSIONS Pg. 250-255 Objectives: 1. Be able to simplify radical expressions 2. Add, subtract, multiply or divide radical expressions accurately 3. Find solutions to problems using radical expressions and future equations STEPS TO SIMPLIFY RADICAL EXPRESSIONS: 3 ______ √ 108 1) Identify factors of the radicand (number under the radical sign) using basic numbers raised to the power of the index (Factors of 108 based on cubed power are 27 (33) times 4) 2) Isolate the perfect factor, then simplify and recombine results 3 ____ 3 ____ √ 108 = (108)1/3 = (27)1/3 times 4(1/3) = 3 times 4(1/3) = 3√ 108 NOTE: When dealing with even numbered index (the small number above) be sure both factors are non-negative! Why? Because the even root of any negative number is an imaginary number. Example 1: Simplify 2 _______ ______________ __ √25a b = √5 (a ) (b ) b = 5 4 9 2 2 2 4 2 2/2 2 2/2 4 2/2 1/2 (a ) (b ) b = 5a b √ b 2 4 THE DIVISION OF RADICAL PROCESS NOTE: Answers must NOT have radicals in the denominator! Therefore, we must rationalize the denominator to eliminate the radical. (In real words, multiply the denominator by another radical (in the form of 1/1 to form a perfect square, cube, etc depending on the index to eliminate the radical on the bottom! Remember this? (1/2)(2/2) = 2/4 We didn’t change the value of the fraction, just the looks. We will use this same principle to eliminate the radical from the denominator of a fraction. ____ ___ __ __ ___ ___ Example 2 √ 7 = √ 7 = √7 times √5 = √35 = √35 5 √5 √5 √5 √25 5 NOTE: Any radical of index “2” called the square root, when multiplied by itself will yield the radicand (the number under the radical.) or in our example the “5” ** Remember to factor the numerator if at all possible. What if the index is a number other than 2? Then we need to determine the value of the radicand to the power of the index. Usually multiply the top and bottom by the radicand to a power that make the resulting denominator power divisible by the index under the radical. This will yield a value of the radicand that will “magically” factor out of the radical. This will require the assistance of an example to illustrate the importance of the changing index. 5____ y8 Example 3 √ = x7 5___ y8 √ √ x7 3__ Try this problem: √2 9x = __ _____ 5__ 5 ___ 8/5 3/5 8 3 8 3 = √ y times √ x = √ y x = y x = y √y3x3 √ x7 √ x3 √ x10 x10/5 x2 5 3__ 2 √6x 3x MULTIPLYING RADICALS 3_____ Think of this as a single number: 5 √100a2 (as one unit together) then multiplied by another single unit (both having the same index) 3 ____ 3 ___ 3 ______ 3 ______ 5√100a2 times √10a = 5√1000a3 = 5√103 a3 = 5(10)(a) = 50a ADDING OR SUBTRACTING RADICALS Note: To either add or subtract radicals, BOTH INDEX and RADICALS must be alike in both terms to combine the results into one unit. __ __ __ __ __ ___ 2 √3 + 3 √3 = 5√3 OR 6√17 - 4√17 = 2√17 MULTIPLYING SUMS OR DIFFERENCES OF RADICALS (FOIL and combine like terms as necessary) __ _ __ (2√3 + 3√5)(3 – √3 ) = F O I L (2√3)(3) + (2√3)(- √3) + (3√5)(3) + (3√5)(- √3) 6√3 + - 2√9 + 9√5 + - 3√15 6√3 + (-2)(3) + 9√5 + (-3)√15 __ __ __ = 6√3 – 6 + 9√5 - 3√15 Rationalize the Denominators using a Conjugate of the Denominator Conjugate: A Binomial of the form (a + b)(a – b) if when “FOIL”ed will eliminate the center term This process, when used with radicals, effectively eliminates the radical by squaring the first and last terms, while the middle two terms cancel each other out (like the Difference of Two Squares) Example 4 2 + √3 times 4 + √3 4 - √3 4 + √3 = 8 + 2√3 + 4√3 +(√3)2 = 8 + 3 + 6√3 = 11 + 6√3 16 + 4√3 – 4√3 - (√3)2 16 – 3 13 NOTE: 11 + 6√3 ≠ 17√3 It’s like adding 11 + 6x with only “like terms” combining!